894 13. MID-SIZE GROUPS OVER F2Next Ca(V3) ::; G2 ::; Na(X), so as NL(V3) normalizes Ca(V3), Ca(V3) acts
on XNL(Vs). Then as L = (XNL(V^3 )), we conclude Ca(V3)::; Na(L) = M. Hence
Ca(V 3 ) ::; Na(V) = Mv as Vis a TI-set in M by 12.2.6. This establishes (6). D
13.3.1. Eliminating U 3 (3). With the technical results from the earlier part
of the section in hand, we are now ready to embark on the main project in this
chapter: the treatment of the cases L ~ U 3 (3), A6, and A5.
In this subsection, we handle the easiest of these cases:
THEOREM 13.3.16. Assume Hypothesis 13.3.1. Then Lis not U3(3).
In the remainder of this section, assume G, L is a counterexample to Theorem
13.3.16. By 13.3.2.3, Cv(L) = 1 and m(V) = 6. In particular Hypothesis 13.3.13
is satisfied, so we can appeal to 13.3.14 and 13.3.15. Let z be a generator for V 1 ,
so that G1 := Ca(z) = Gz, and Ch = G1/V1. As usual d~fine
Hz= {HE H(L1T): H ~ G1 and Hi. M}.
By 13.3.6, G1 i. M, so G1 E Hz and hence Hz-=/= 0.
We first observe:
LEMMA 13.3.17. (1) G1 n G3::; Mv?: Ca(V3).
(2) r(G, V) > 3.
(3) If [V, Vg] ::; V n Vg, then [V, Vg] = 1.
(4) [02(G1), Vz]-=/= 1.PROOF. Part (4) holds by 13.3.14, and Ca(V3) ::; Mv by 13.3.15.6. Further
AutM 1 (V3) is the full stabilizer in GL(V3) of Vi, so G1 n G3 = Ca(V3)NM 1 (V3). As
V is a TI-set in M by 12.2.6, this completes the proof of (1). Then (1) together
with parts (2) and (3) of 13.3.12 imply (2), while (1) and part (4) of 13.3.12 imply
~- D
LEMMA 13.3.18. (1) For each HE Hz, Hypothesis F.9.1 is satisfied with V3 in
the roles of ''V+ ".(2) (V^01 ) is abelian.
PROOF. We check the various parts of Hypothesis F.9.1:
First hypothesis (c) of F.9.1 follows from 13.3.17.1, and by construction L 1 is
irreducible on V 3 , so hypot.hesis (b) holds. As H E H(T), H E He by 1.1.4.6.
Also by Coprime Action and 13.3.17.1, Y := 02 (CH(i/ 3 )) ::; CMv(V3)), so as
02 (C.Mv(V3)) = 1, Y ~ CM(V) ::; CM(L/0 2 (L)) and therefore L normalizes Y =
02 (Y0 2 (L)). Thus if Y -=/= 1, then H ::; Na(Y) ::; M = !M(LT), contrary to
the definition of H E Hz- Thus CH(V 3 ) is a 2-group, so hypothesis (a) follows.
As M = !M(LT) and H i. M, hypothesis (d) holds. Finally 13.3.17.3 implies
hypothesis ( e), completing the proof of (1).
Now let H := G1, and as in Hypothesis F.9.1, define UH:= (V 3 H), VH := (VH),
QH = 02(H) and H* := H/CH(UH)· It remains to prove (2), so we may assume
VH is nonabelian.